Mr Jenkins, 61, was the oldest victim of the disaster.
The humanist service was led by Ken Eadie and celebrated Mr Jenkins' love of music.
Mourners entered the crematorium to the strains of Wonderful Land by The Shadows.
The retired Scottish Gas administration worker had two daughters, Claire, 32, and Elaine, 31, with his wife of more than 30 years, Margaret, a school teacher who died in 2006.
He was a keen runner and had appeared as an extra in the BBC dramas Fields Of Blood, and Single Father, which starred David Tennant, and also BBC soap River City.
As well as his daughters, among the mourners were his partner of four years, Mary Kavanagh, and his sister Fiona Johnstone.
Mr Jenkins was listening to the band in the Clutha with Ms Kavanagh when the helicopter hit the building. She escaped the bar unharmed and was restrained from going back in to find Mr Jenkins, known to friends as Rab.
Lord Provost Sadie Docherty and Glasgow City Council Gordon Matheson were among those at the service.
A special poem about courage was read by Ms Kavanagh, while two life-long friends Tony Darroch and Davy Welsh paid tribute to Mr Jenkins, who grew up in Balornock, in the city's north east.
In a statement released ahead of the funeral, his family said: "Robert Jenkins was a loving father and a devoted brother.
"Robert was an especially kind, warm and funny man with many friends … he will be greatly missed by all who knew him."
A recording of Auld Lang Syne performed by Mairi Campbell was played at the memorial and mourners left the building to the sound of the Beatles' Eight Days A Week.
A message on the Order Of Service thanked all relatives and friendss for "their overwhelming kindness, love and support" and also thanked the emergency services.